344 XV NORMED ALGEBRAS AND SPECTRAL THEORY (d) If a, b e A are self-adjoint, show that p(a2) <£ p(a2 -f b2). (Remark that p(a2 + b2)e - a2 = (p(a2 + b2)e -a2- b2) + b2, and use (c).) Deduce that p(x* + x) <; 2/>(x) for all * e A. (e) Deduce from (b) and (d) that p(x + y) ^p(x) +p(y) for all x, y E A. If A is without radical, it follows that p defines on A a normed algebra structure, for which the topology is coarser than that defined by the norm \\x\\. These two topologies coincide only if A is complete relative to the norm p. (f) Show that if A is without radical, the function x*-+x* is continuous on A. (Use the closed graph theorem, by remarking that if a sequence (xn) is such that xn -» 0 and x% -+y, then p(x*) and p(y — x*) both tend to 0; then use (e).) (g) Show that p is a continuous function on A. (Use (f), by remarking that there exists a constant c>0 such that ||TT(;C*)|| ^c|br(jc)||, and that pA(x) = pA/ttW*)).) (h) Show that, for each x e A, SpOt**) is contained in [0, + oo[. (Argue by contra- diction: suppose that there exists x e A such that — 1 e Sp(x*x). Write z — x*x, and for each positive integer n let wn be a self-adjoint element of A which commutes with z, and is such that w% = z2 + - e and Sp(wn) <= [0, -j- oo [ (Problem 17). Let bn = wn — z. n By considering a commutative Banach subalgebra B of A containing wn and z, and such that the spectra of z, wn , and bn are the same in B and in A, show that Show also that p(bn) <* 1 + 2p(z) = a, independent of n (use (c)). Put yn — xbn , so that y*yn = — b2nwn + Wn)bn; deduce that Sp(y$yn) is contained in the interval ]— oo, a//i], and then that Sp(jw y%) is contained in the interval [— a///, + oo[ (use (c)); hence that p(y%yn) ^ ot/n (Section 1 5.2, Problem 2(b)), and consequently that IxMjO! ^oc/n for all characters x °f B- Finally, obtain a contradiction by noting that ylyn = b2z, that x(wn) ^ 0 and that there exists a character x such that x(z) = — 1 .) 19. Let A be a Banach algebra with unit element, and let xt~+x* be a (not necessarily continuous) involution on A. Show that the following properties of A are equivalent : (a) A is hermitian. (j8) p(x) ^p(x) for all normal x e A. (y) p(x) — p(x) for all normal x e A. (S) p(** + *)g2X*)forallx6A. (e) p(x + y) <p(x) + Xy) for all x, y e A. (£) The set of numbers p(x), where * runs through the unitary elements of A, is bounded. (r)) For each x e A the element e + x*x is invertible in A. (To show that (17) implies (a), argue by contradiction by showing that the spectrum of a hermitian element cannot contain the number /. To show that (y) implies (a), argue as in (15.4.12). To show that (8) implies (/3), remark that (8) implies that p(x)n = p(xn) g 2p(x)n for x normal and n ;> 1 . Finally, to show that (£) implies (a), notice first that (£) implies that Sp(;c) c U for all unitary elements x, by considering the powers xn (n e Z). Next observe that if a is self-adjoint and p(a) < I , there exists a self-adjoint element b, commuting with a, such that b2 = <? — a2 (Problem 17), hence a + & is unitary. Then consider a commutative Banach subalgebra containing a and 6 and such that the spectra of a, b and a + ib are the same in A and B).ments cz, b of A,